Noninvasive Spatial Metrology of Single-Atom Devices
Purdue University
Purdue e-Pubs Birck and NCN Publications
Birck Nanotechnology Center
5-2013
Noninvasive Spatial Metrology of Single-Atom Devices Fahd A. Mohiyaddin University of New South Wales
Rajib Rahman Sandia National Laboratory
Rachpon Kalra University of New South Wales
Gerhard Klimeck Network for Computational Nanotechnology, Birck Nanotechnology Center, Purdue University, [email protected]
Lloyd C. L. Hollenberg University of Melbourne See next page for additional authors
Follow this and additional works at: http://docs.lib.purdue.edu/nanopub Part of the Nanoscience and Nanotechnology Commons Mohiyaddin, Fahd A.; Rahman, Rajib; Kalra, Rachpon; Klimeck, Gerhard; Hollenberg, Lloyd C. L.; Pla, Jarryd J.; Dzurak, Andrew S.; and Morello, Andrea, "Noninvasive Spatial Metrology of Single-Atom Devices" (2013). Birck and NCN Publications. Paper 1401. http://docs.lib.purdue.edu/nanopub/1401
This document has been made available through Purdue e-Pubs, a service of the Purdue University Libraries. Please contact [email protected] for additional information.
Authors
Fahd A. Mohiyaddin, Rajib Rahman, Rachpon Kalra, Gerhard Klimeck, Lloyd C. L. Hollenberg, Jarryd J. Pla, Andrew S. Dzurak, and Andrea Morello
This article is available at Purdue e-Pubs: http://docs.lib.purdue.edu/nanopub/1401
Letter pubs.acs.org/NanoLett
Noninvasive Spatial Metrology of Single-Atom Devices Fahd A. Mohiyaddin,*,† Rajib Rahman,‡,⊥ Rachpon Kalra,† Gerhard Klimeck,§ Lloyd C. L. Hollenberg,∥ Jarryd J. Pla,† Andrew S. Dzurak,† and Andrea Morello*,† †
Centre for Quantum Computation and Communication Technology, School of Electrical Engineering and Telecommunications, University of New South Wales, Sydney NSW 2052, Australia ‡ Advanced Device Technologies, Sandia National Laboratories, Albuquerque, New Mexico 87185, United States § Network for Computational Nanotechnology, Purdue University, West Lafayette, Indiana 47907, United States ∥ Centre for Quantum Computation and Communication Technology, School of Physics, University of Melbourne, Melbourne VIC 3010, Australia S Supporting Information *
ABSTRACT: The exact location of a single dopant atom in a nanostructure can influence or fully determine the functionality of highly scaled transistors or spin-based devices. We demonstrate here a noninvasive spatial metrology technique, based on the microscopic modeling of three electrical measurements on a single-atom (phosphorus in silicon) spin qubit device: hyperfine coupling, ground state energy, and capacitive coupling to nearby gates. This technique allows us to locate the qubit atom with a precision of ±2.5 nm in two directions and ±15 nm in the third direction, which represents a 1500-fold improvement with respect to the prefabrication statistics obtainable from the ion implantation parameters. KEYWORDS: triangulation
31
P donors in Si, quantum computing, ion implantation, donor location uncertainty, nanoelectronic modeling,
T
position of the atom.20 While this can be mitigated in part by using very low implantation energies, it is generally the case that only a probabilistic description of the final atom position can be given, using Monte Carlo techniques.21 Therefore, techniques that accurately reveal (postfabrication) the position of an implanted dopant are crucial to inform further developments in single-atom nanoelectronic devices. Scanning probe techniques can sometimes be used to accurately locate a dopant,1 but most often it is necessary to resort to invasive methods such as atom-probe tomography,22 which destroys a fully functional device in the process. Moreover, in the presence of multiple dopants, the above techniques do not allow to identify which specific dopant is responsible for the observed single-atom effects. This calls for a noninvasive technique that can successfully pinpoint the location of dopants that influence the device behavior, and hence provide real-time feedback to an experiment. Here, we present a novel modeling protocol that allows us to locate a deliberately implanted P donor in a silicon MOS spin qubit device.23 Our non-invasive spatial metrology method enables a triangulation of the donor position which narrows the
he ability to place and control individual dopant atoms in semiconductor nanostructures is paving the way to exciting and novel device functionalities.1 For classical but highly miniaturized electronic devices, it is well established that the number and location of individual dopants has tangible effects on the performance of ultrasmall silicon transistors.2−4 The pursuit of “deterministic doping” constitutes an important part of the effort toward further scaling of silicon chips.5 At the extreme limit where a single dopant atom determines the device functionality, it is now possible to realize “single-atom transistors”.6−8 Beyond classical nanoelectronic devices, single dopant atoms in silicon have been proposed as an excellent physical platform to encode and manipulate quantum information, either in the spin9−12 or in the charge13 degree of freedom. Indeed, the single-shot readout14 and the coherent control15 of a singleatom spin qubit in a silicon metal−oxide−semiconductor (MOS) structure have been recently demonstrated, as well as the control of the charge states of donor pairs.16,17 While great progress is being made on the atomically precise positioning of single atoms by scanning probe techniques,8,18 ion implantation19 remains the leading method to introduce dopants in semiconductor nanostructures in industry as well as in basic research. The inherent limitation of ion-implanted devices at the single-atom level is the straggle in the final © 2013 American Chemical Society
Received: October 19, 2012 Revised: March 18, 2013 Published: April 9, 2013 1903
dx.doi.org/10.1021/nl303863s | Nano Lett. 2013, 13, 1903−1909
Nano Letters
Letter
Figure 1. (a) Schematic of the spin qubit device. The electrostatic gates induce and confine electrons underneath the Si/SiO2 interface. The electron density indicated in red-yellow scale highlights the position of the SET island with respect to the implant region and metallic gates. (b) Conduction band profile for the spin readout criterion, where the donor ground state is tuned in resonance with the Fermi level of SET island. (c) Charge stability diagram with transitions arising from electron tunneling between the donor and the SET island. The experimental values of voltages on the top and plunger gates shown in the figure, as well the voltages on the left and right barrier gates (0.65 V), were used to calculate the potential energy and electric field profiles. (d) Electron spin resonance spectra, showing two resonance peaks separated by the contact hyperfine splitting.
possible locations to a volume 1500 times smaller than what could have been predicted by prefabrication statistical methods. The triangulation is obtained by combining three different techniques, each one predicting a locus of donor locations compatible with a measurable physical property of the system: (i) A classical, finite-elements electrostatic simulation to match the ground state energy of the donor-bound electron; (ii) A quantum-mechanical tight-binding model24,25 for the wave function of the donor-bound electron to match the Stark-shift of the hyperfine coupling;26,27 (iii) A geometric capacitance extraction method28 to match the measured capacitive coupling between the donor and the surrounding electrodes. The schematic of the experimental device is displayed in Figure 1a. A single-electron transistor (SET) is formed under the Si/SiO2 interface, by appropriate gate biasing of the metallic gates. The qubit is an individual phosphorus donor with both capacitive and tunnel coupling to the SET island. The donor under study is one of many (∼16 maximum likelihood, from Poisson statistics) P atoms implanted in a 90 × 90 nm2
window. For electron spin readout, the donor electron ground state energy is tuned in resonance with the Fermi Level of the SET island (referred to as “spin readout criterion”), as shown in Figure 1b. In an externally applied magnetic field B0, the Zeeman splitting between the spin states results in preferential tunneling14 of the donor spin-up electron into the island. The SET is tuned such that the spin-dependent donor ionization results in a large current, whereas no current flows while the donor is neutral (Coulomb blockade). For B0 = 0, charge transitions are observed in the charge stability diagram (Figure 1c), arising from tunneling of the donor electron to the island. The transitions are characterized by the charge transfer signal (CTS)23 Δq/e = V2/V1 = 0.42, where V1 is the SET Coulomb spacing, and V2 is the shift in the Coulomb peaks caused by donor ionization. Δq/e also corresponds to the ratio of capacitances Cm/CΣ, where Cm is the mutual capacitance between the donor and the island, while CΣ is the total capacitance of the donor, including its self-capacitance and mutual capacitances to various gates and the SET island. 1904
dx.doi.org/10.1021/nl303863s | Nano Lett. 2013, 13, 1903−1909
Nano Letters
Letter
A microwave transmission line fabricated on the chip29 (not included in Figure 1a) generates an oscillating magnetic field B1 which enables electron spin control. Electron spin resonance (ESR) measurement results are plotted in Figure 1d and show two resonant peaks dependent on the state of the 31P nuclear spin. The two peaks are separated by the contact hyperfine splitting (HFS), which takes the value of 4.086 mT in the experiment.30 Because of the strong electric fields in the nanostructure, this value is Stark-shifted from the value of 4.2 mT observed in bulk samples.31 In the remainder of this paper, we will sequentially address the simulation strategy and results of the hyperfine splitting, the spin readout criterion, the charge transfer signal, and their triangulation. The spin Hamiltonian of a donor electron spin S and nuclear spin I in an applied electric field ε and magnetic field B0 is given by: H = ge(ε)μB S·B0 − gnμn I·B0 + A(ε)I·S
|ψ (r0 , ε)|2 A (ε ) = A(0) |ψ (r0 , 0)|2
(2)
The electric field dependence of the HFS for a bulk donor was investigated previously using uniform fields in tightbinding,27 and agreed well with ESR measurements on bulk donor ensembles.26 However, the calculation of the HFS in a nanoscale device must account for significant variations of the electric field within the donor Bohr radius, arising from the gates and ionized donors, and also due to proximal heterointerfaces. Hence, a realistic potential and electric field profile from TCAD is essential to model the experiments. The magnitude of the electric field along a device slice is shown in Figure 2a. The inset of Figure 2a shows the ratio of
(1)
The first and second terms in eq 1 are electronic and nuclear Zeeman terms, the third term is the contact hyperfine interaction between the two spins, ge(ε)32 and gn are the electron and nuclear gyromagnetic ratios, and μB and μn are the Bohr and nuclear magnetons, respectively. The contact hyperfine interaction is expressed as A(ε) = (8π/3)gegnμBμn|ψ(r0,ε)|2, where |ψ(r0,ε)|2 is the probability density of the electron wave function evaluated at the donor site r0. In realistic devices, |ψ(r0,ε)|2 is distorted from the bulk value |ψ(r0,0)|2, resulting in a Stark shift of the HFS. Hence, to electrostatically model the HFS, it is essential to obtain the potential profile in the nanostructure and hence the electric fields, to be able to estimate the distortion of the donor electron wave function. The software ISE-TCAD,33 a finite-element Poisson equation solver, is used to obtain the potential profile taking into account the exact geometry of the device. In addition, the simulated threshold voltage in the model was calibrated to the experimental value of 0.7 V, by addition of interface charges with charge density Qox = −3.4 × 1011 cm−2 at the Si/SiO2 boundary. This type of a-posteriori estimate of the fixed interface charge density is well established in the modeling of MOSFET structures. The calculated charge density is also consistent with estimates from deep level transient spectroscopic measurements.34 The model also captures the effect of ionized donor potentials, by including a uniform positive charge density (4.63 × 1016 cm−3) corresponding to the remaining 15 ionized donors in the implant region. We have also performed analysis for scenarios without ionized donors. The shift in donor locations, arising from the presence of ionized donors is by about 5 nm in the y-direction, and is comparable to the resulting donor location uncertainty obtained after modeling (discussed further below). This necessitates the inclusion of ionized donors in the TCAD model. The donor is modeled as a Coulomb potential superimposed on the TCAD potential profile, with an on-site truncation value calibrated27,35 to obtain the bulk binding energy (45.6 meV) for 31P donors in silicon. The wave functions for the complete potential profile are then obtained by solving the full atomistic tight-binding Hamiltonian with Nano Electronic MOdeling 3D (NEMO 3D) tool.24,25 The tight-binding model used here employs a 20-band sp3d5s* nearest neighbor Hamiltonian, where wave functions are expressed with a linear combination of atomic orbitals. The hyperfine splitting relative to the bulk value is given by27,36
Figure 2. (a) Electric fields in a device cross section (yz-plane) cut through the center of the plunger gate. Inset: Electric-field-induced distortion of the electron probability density for a donor positioned at a specific lattice site r0 = (0,7.96,−9.86). (b) Hyperfine splitting, Starkshifted from the bulk value, along the yz-plane. The dashed green line represents the amount by which the SET island extends beyond the lithographic dimension of the top gate. The dotted gray line represents the approximate boundary of the implant region in the y- and zdirections.
electron probability densities for a donor located at a specific lattice site r0, and for a bulk donor. As a consequence of the direction of the electric fields in the vicinity of the donor, the electron probability density is not only pushed toward the direction of the SET island, but also away from the Si/SiO2 interface. Noticeable probability density distortion for the donor at r0 is apparent from the inset, and results in a significant Stark shift of the contact HFS. 1905
dx.doi.org/10.1021/nl303863s | Nano Lett. 2013, 13, 1903−1909
Nano Letters
Letter
Figure 2b depicts the contact term of the HFS for various donor positions r0 along the device slice. This exploration required about 600 independent atomistic NEMO-3D calculations for as many different donor positions. The map of HFS can be divided into three regions, corresponding to deep (z < −25 nm), intermediate depth (−25 nm < z < −5 nm) and very shallow (z > −5 nm) donors. The Stark shift is small for deep donors, with HFS close to the bulk value in electric fields lower than 1 MV/m. For intermediate depth donors, the electron probability density at r0 is severely reduced by the large electric field, resulting in hyperfine splittings as small as 2.8 mT. The proximity of the Si/SiO2 interface to the donor also influences HFS in addition to the electric field. The abrupt potential barrier modifies the wave function confinement, and hence different HFS are noticed for donors along iso-electric field lines. Additionally, the potential barrier tends to confine the electron probability density more than the bulk, thereby allowing values of the HFS beyond 4.2 mT for very shallow donors, despite the large electric field. This increase has been previously reported and analyzed with the help of firstorder perturbation theory.27 As a consequence of the variation of local electric fields and the presence of the Si/SiO2 interface, the locus of donor locations compatible with the experimental HFS value of 4.086 mT takes the shape of the irregular semiellipsoid represented by the white contour in Figure 2b. To simulate the spin readout criterion, we map out the potential energy of the system, for the set of gate voltages applied in experiment. TCAD serves this purpose by computing the conduction band profile, Ec. The TCAD map of Ec, with the Fermi level EF set as the reference zero energy, is plotted in Figure 3a. The conduction band tends to rise in energy toward the direction of the plunger gate, since the top and plunger gates are biased at high and low positive gate voltages, respectively. The SET island extends slightly beyond the lithographic dimensions of the top gate, due to the lowering of conduction band below the Fermi level (green region in Figure 3a). By integrating the electron density in Figure 1a, 93 electrons were estimated to be present in the SET island during spin readout. The white contour in Figure 3a denotes locations where the TCAD conduction band energy is equal to the bulk binding energy of the donor. We assume that the spin readout criterion is fulfilled at these locations, since the donor ground state E0 aligns close to EF. The validity of the above assumption rests on the observation that the variation of Ec in Figure 3a is much larger than the modification of donor ground state binding energy by Stark shift and hybridization with the island (
Purdue e-Pubs Birck and NCN Publications
Birck Nanotechnology Center
5-2013
Noninvasive Spatial Metrology of Single-Atom Devices Fahd A. Mohiyaddin University of New South Wales
Rajib Rahman Sandia National Laboratory
Rachpon Kalra University of New South Wales
Gerhard Klimeck Network for Computational Nanotechnology, Birck Nanotechnology Center, Purdue University, [email protected]
Lloyd C. L. Hollenberg University of Melbourne See next page for additional authors
Follow this and additional works at: http://docs.lib.purdue.edu/nanopub Part of the Nanoscience and Nanotechnology Commons Mohiyaddin, Fahd A.; Rahman, Rajib; Kalra, Rachpon; Klimeck, Gerhard; Hollenberg, Lloyd C. L.; Pla, Jarryd J.; Dzurak, Andrew S.; and Morello, Andrea, "Noninvasive Spatial Metrology of Single-Atom Devices" (2013). Birck and NCN Publications. Paper 1401. http://docs.lib.purdue.edu/nanopub/1401
This document has been made available through Purdue e-Pubs, a service of the Purdue University Libraries. Please contact [email protected] for additional information.
Authors
Fahd A. Mohiyaddin, Rajib Rahman, Rachpon Kalra, Gerhard Klimeck, Lloyd C. L. Hollenberg, Jarryd J. Pla, Andrew S. Dzurak, and Andrea Morello
This article is available at Purdue e-Pubs: http://docs.lib.purdue.edu/nanopub/1401
Letter pubs.acs.org/NanoLett
Noninvasive Spatial Metrology of Single-Atom Devices Fahd A. Mohiyaddin,*,† Rajib Rahman,‡,⊥ Rachpon Kalra,† Gerhard Klimeck,§ Lloyd C. L. Hollenberg,∥ Jarryd J. Pla,† Andrew S. Dzurak,† and Andrea Morello*,† †
Centre for Quantum Computation and Communication Technology, School of Electrical Engineering and Telecommunications, University of New South Wales, Sydney NSW 2052, Australia ‡ Advanced Device Technologies, Sandia National Laboratories, Albuquerque, New Mexico 87185, United States § Network for Computational Nanotechnology, Purdue University, West Lafayette, Indiana 47907, United States ∥ Centre for Quantum Computation and Communication Technology, School of Physics, University of Melbourne, Melbourne VIC 3010, Australia S Supporting Information *
ABSTRACT: The exact location of a single dopant atom in a nanostructure can influence or fully determine the functionality of highly scaled transistors or spin-based devices. We demonstrate here a noninvasive spatial metrology technique, based on the microscopic modeling of three electrical measurements on a single-atom (phosphorus in silicon) spin qubit device: hyperfine coupling, ground state energy, and capacitive coupling to nearby gates. This technique allows us to locate the qubit atom with a precision of ±2.5 nm in two directions and ±15 nm in the third direction, which represents a 1500-fold improvement with respect to the prefabrication statistics obtainable from the ion implantation parameters. KEYWORDS: triangulation
31
P donors in Si, quantum computing, ion implantation, donor location uncertainty, nanoelectronic modeling,
T
position of the atom.20 While this can be mitigated in part by using very low implantation energies, it is generally the case that only a probabilistic description of the final atom position can be given, using Monte Carlo techniques.21 Therefore, techniques that accurately reveal (postfabrication) the position of an implanted dopant are crucial to inform further developments in single-atom nanoelectronic devices. Scanning probe techniques can sometimes be used to accurately locate a dopant,1 but most often it is necessary to resort to invasive methods such as atom-probe tomography,22 which destroys a fully functional device in the process. Moreover, in the presence of multiple dopants, the above techniques do not allow to identify which specific dopant is responsible for the observed single-atom effects. This calls for a noninvasive technique that can successfully pinpoint the location of dopants that influence the device behavior, and hence provide real-time feedback to an experiment. Here, we present a novel modeling protocol that allows us to locate a deliberately implanted P donor in a silicon MOS spin qubit device.23 Our non-invasive spatial metrology method enables a triangulation of the donor position which narrows the
he ability to place and control individual dopant atoms in semiconductor nanostructures is paving the way to exciting and novel device functionalities.1 For classical but highly miniaturized electronic devices, it is well established that the number and location of individual dopants has tangible effects on the performance of ultrasmall silicon transistors.2−4 The pursuit of “deterministic doping” constitutes an important part of the effort toward further scaling of silicon chips.5 At the extreme limit where a single dopant atom determines the device functionality, it is now possible to realize “single-atom transistors”.6−8 Beyond classical nanoelectronic devices, single dopant atoms in silicon have been proposed as an excellent physical platform to encode and manipulate quantum information, either in the spin9−12 or in the charge13 degree of freedom. Indeed, the single-shot readout14 and the coherent control15 of a singleatom spin qubit in a silicon metal−oxide−semiconductor (MOS) structure have been recently demonstrated, as well as the control of the charge states of donor pairs.16,17 While great progress is being made on the atomically precise positioning of single atoms by scanning probe techniques,8,18 ion implantation19 remains the leading method to introduce dopants in semiconductor nanostructures in industry as well as in basic research. The inherent limitation of ion-implanted devices at the single-atom level is the straggle in the final © 2013 American Chemical Society
Received: October 19, 2012 Revised: March 18, 2013 Published: April 9, 2013 1903
dx.doi.org/10.1021/nl303863s | Nano Lett. 2013, 13, 1903−1909
Nano Letters
Letter
Figure 1. (a) Schematic of the spin qubit device. The electrostatic gates induce and confine electrons underneath the Si/SiO2 interface. The electron density indicated in red-yellow scale highlights the position of the SET island with respect to the implant region and metallic gates. (b) Conduction band profile for the spin readout criterion, where the donor ground state is tuned in resonance with the Fermi level of SET island. (c) Charge stability diagram with transitions arising from electron tunneling between the donor and the SET island. The experimental values of voltages on the top and plunger gates shown in the figure, as well the voltages on the left and right barrier gates (0.65 V), were used to calculate the potential energy and electric field profiles. (d) Electron spin resonance spectra, showing two resonance peaks separated by the contact hyperfine splitting.
possible locations to a volume 1500 times smaller than what could have been predicted by prefabrication statistical methods. The triangulation is obtained by combining three different techniques, each one predicting a locus of donor locations compatible with a measurable physical property of the system: (i) A classical, finite-elements electrostatic simulation to match the ground state energy of the donor-bound electron; (ii) A quantum-mechanical tight-binding model24,25 for the wave function of the donor-bound electron to match the Stark-shift of the hyperfine coupling;26,27 (iii) A geometric capacitance extraction method28 to match the measured capacitive coupling between the donor and the surrounding electrodes. The schematic of the experimental device is displayed in Figure 1a. A single-electron transistor (SET) is formed under the Si/SiO2 interface, by appropriate gate biasing of the metallic gates. The qubit is an individual phosphorus donor with both capacitive and tunnel coupling to the SET island. The donor under study is one of many (∼16 maximum likelihood, from Poisson statistics) P atoms implanted in a 90 × 90 nm2
window. For electron spin readout, the donor electron ground state energy is tuned in resonance with the Fermi Level of the SET island (referred to as “spin readout criterion”), as shown in Figure 1b. In an externally applied magnetic field B0, the Zeeman splitting between the spin states results in preferential tunneling14 of the donor spin-up electron into the island. The SET is tuned such that the spin-dependent donor ionization results in a large current, whereas no current flows while the donor is neutral (Coulomb blockade). For B0 = 0, charge transitions are observed in the charge stability diagram (Figure 1c), arising from tunneling of the donor electron to the island. The transitions are characterized by the charge transfer signal (CTS)23 Δq/e = V2/V1 = 0.42, where V1 is the SET Coulomb spacing, and V2 is the shift in the Coulomb peaks caused by donor ionization. Δq/e also corresponds to the ratio of capacitances Cm/CΣ, where Cm is the mutual capacitance between the donor and the island, while CΣ is the total capacitance of the donor, including its self-capacitance and mutual capacitances to various gates and the SET island. 1904
dx.doi.org/10.1021/nl303863s | Nano Lett. 2013, 13, 1903−1909
Nano Letters
Letter
A microwave transmission line fabricated on the chip29 (not included in Figure 1a) generates an oscillating magnetic field B1 which enables electron spin control. Electron spin resonance (ESR) measurement results are plotted in Figure 1d and show two resonant peaks dependent on the state of the 31P nuclear spin. The two peaks are separated by the contact hyperfine splitting (HFS), which takes the value of 4.086 mT in the experiment.30 Because of the strong electric fields in the nanostructure, this value is Stark-shifted from the value of 4.2 mT observed in bulk samples.31 In the remainder of this paper, we will sequentially address the simulation strategy and results of the hyperfine splitting, the spin readout criterion, the charge transfer signal, and their triangulation. The spin Hamiltonian of a donor electron spin S and nuclear spin I in an applied electric field ε and magnetic field B0 is given by: H = ge(ε)μB S·B0 − gnμn I·B0 + A(ε)I·S
|ψ (r0 , ε)|2 A (ε ) = A(0) |ψ (r0 , 0)|2
(2)
The electric field dependence of the HFS for a bulk donor was investigated previously using uniform fields in tightbinding,27 and agreed well with ESR measurements on bulk donor ensembles.26 However, the calculation of the HFS in a nanoscale device must account for significant variations of the electric field within the donor Bohr radius, arising from the gates and ionized donors, and also due to proximal heterointerfaces. Hence, a realistic potential and electric field profile from TCAD is essential to model the experiments. The magnitude of the electric field along a device slice is shown in Figure 2a. The inset of Figure 2a shows the ratio of
(1)
The first and second terms in eq 1 are electronic and nuclear Zeeman terms, the third term is the contact hyperfine interaction between the two spins, ge(ε)32 and gn are the electron and nuclear gyromagnetic ratios, and μB and μn are the Bohr and nuclear magnetons, respectively. The contact hyperfine interaction is expressed as A(ε) = (8π/3)gegnμBμn|ψ(r0,ε)|2, where |ψ(r0,ε)|2 is the probability density of the electron wave function evaluated at the donor site r0. In realistic devices, |ψ(r0,ε)|2 is distorted from the bulk value |ψ(r0,0)|2, resulting in a Stark shift of the HFS. Hence, to electrostatically model the HFS, it is essential to obtain the potential profile in the nanostructure and hence the electric fields, to be able to estimate the distortion of the donor electron wave function. The software ISE-TCAD,33 a finite-element Poisson equation solver, is used to obtain the potential profile taking into account the exact geometry of the device. In addition, the simulated threshold voltage in the model was calibrated to the experimental value of 0.7 V, by addition of interface charges with charge density Qox = −3.4 × 1011 cm−2 at the Si/SiO2 boundary. This type of a-posteriori estimate of the fixed interface charge density is well established in the modeling of MOSFET structures. The calculated charge density is also consistent with estimates from deep level transient spectroscopic measurements.34 The model also captures the effect of ionized donor potentials, by including a uniform positive charge density (4.63 × 1016 cm−3) corresponding to the remaining 15 ionized donors in the implant region. We have also performed analysis for scenarios without ionized donors. The shift in donor locations, arising from the presence of ionized donors is by about 5 nm in the y-direction, and is comparable to the resulting donor location uncertainty obtained after modeling (discussed further below). This necessitates the inclusion of ionized donors in the TCAD model. The donor is modeled as a Coulomb potential superimposed on the TCAD potential profile, with an on-site truncation value calibrated27,35 to obtain the bulk binding energy (45.6 meV) for 31P donors in silicon. The wave functions for the complete potential profile are then obtained by solving the full atomistic tight-binding Hamiltonian with Nano Electronic MOdeling 3D (NEMO 3D) tool.24,25 The tight-binding model used here employs a 20-band sp3d5s* nearest neighbor Hamiltonian, where wave functions are expressed with a linear combination of atomic orbitals. The hyperfine splitting relative to the bulk value is given by27,36
Figure 2. (a) Electric fields in a device cross section (yz-plane) cut through the center of the plunger gate. Inset: Electric-field-induced distortion of the electron probability density for a donor positioned at a specific lattice site r0 = (0,7.96,−9.86). (b) Hyperfine splitting, Starkshifted from the bulk value, along the yz-plane. The dashed green line represents the amount by which the SET island extends beyond the lithographic dimension of the top gate. The dotted gray line represents the approximate boundary of the implant region in the y- and zdirections.
electron probability densities for a donor located at a specific lattice site r0, and for a bulk donor. As a consequence of the direction of the electric fields in the vicinity of the donor, the electron probability density is not only pushed toward the direction of the SET island, but also away from the Si/SiO2 interface. Noticeable probability density distortion for the donor at r0 is apparent from the inset, and results in a significant Stark shift of the contact HFS. 1905
dx.doi.org/10.1021/nl303863s | Nano Lett. 2013, 13, 1903−1909
Nano Letters
Letter
Figure 2b depicts the contact term of the HFS for various donor positions r0 along the device slice. This exploration required about 600 independent atomistic NEMO-3D calculations for as many different donor positions. The map of HFS can be divided into three regions, corresponding to deep (z < −25 nm), intermediate depth (−25 nm < z < −5 nm) and very shallow (z > −5 nm) donors. The Stark shift is small for deep donors, with HFS close to the bulk value in electric fields lower than 1 MV/m. For intermediate depth donors, the electron probability density at r0 is severely reduced by the large electric field, resulting in hyperfine splittings as small as 2.8 mT. The proximity of the Si/SiO2 interface to the donor also influences HFS in addition to the electric field. The abrupt potential barrier modifies the wave function confinement, and hence different HFS are noticed for donors along iso-electric field lines. Additionally, the potential barrier tends to confine the electron probability density more than the bulk, thereby allowing values of the HFS beyond 4.2 mT for very shallow donors, despite the large electric field. This increase has been previously reported and analyzed with the help of firstorder perturbation theory.27 As a consequence of the variation of local electric fields and the presence of the Si/SiO2 interface, the locus of donor locations compatible with the experimental HFS value of 4.086 mT takes the shape of the irregular semiellipsoid represented by the white contour in Figure 2b. To simulate the spin readout criterion, we map out the potential energy of the system, for the set of gate voltages applied in experiment. TCAD serves this purpose by computing the conduction band profile, Ec. The TCAD map of Ec, with the Fermi level EF set as the reference zero energy, is plotted in Figure 3a. The conduction band tends to rise in energy toward the direction of the plunger gate, since the top and plunger gates are biased at high and low positive gate voltages, respectively. The SET island extends slightly beyond the lithographic dimensions of the top gate, due to the lowering of conduction band below the Fermi level (green region in Figure 3a). By integrating the electron density in Figure 1a, 93 electrons were estimated to be present in the SET island during spin readout. The white contour in Figure 3a denotes locations where the TCAD conduction band energy is equal to the bulk binding energy of the donor. We assume that the spin readout criterion is fulfilled at these locations, since the donor ground state E0 aligns close to EF. The validity of the above assumption rests on the observation that the variation of Ec in Figure 3a is much larger than the modification of donor ground state binding energy by Stark shift and hybridization with the island (
